Lower Bound:

• For call options, the lower bound is zero. The call option value cannot be negative.

• For put options, the lower bound is also zero. The put option value cannot be negative.

These bounds are important in option pricing, as they provide a range within which the option value must fall. Option pricing models, such as the Black-Scholes model, take these bounds into account to estimate the option value.

While these maximum limits have to be explained separately for European and American options, we will first look into the upper and lower bounds of European call options. 

Upper bounds of European call values:

The value of a European call option is capped at the value of the underlying stock. If the stock price is $60 and the call option value is $65, selling the stock and writing the call yields a $5 profit. The call value at expiry cannot exceed the stock value.

If a $5 dividend is declared, the stock price on the expiry date falls to $55 ($60 - $5). Then, the call value cannot exceed $55.

In summary, the upper bound of a European call is the stock price, or the stock price minus the dividend amount if a dividend is declared.

The upper bound of a European call option is the underlying stock price, and if a dividend is declared, the upper bound is the stock price minus the dividend amount.

To summarize:

The upper bound of the European call option = Min (Stock price, Stock price - Dividend amount)

In the example you gave:

• Without dividend: Upper bound = Stock price = 60

• With dividend of 5: Upper bound = Stock price - Dividend = 60 - 5 = 55

This means that the value of the European call option cannot exceed 55 on expiry, considering the dividend announcement.

Lower bounds of European call values:

The lowest value of a European call option is zero, which occurs when the stock value falls to zero.

In general, the call value cannot fall below the stock value minus the present value of the strike price. This is because one can buy the call, sell the stock, and invest the proceeds in risk-free bonds to earn a profit.

For example, consider a stock trading at Rs. 102 with a European call option at a strike price of Rs. 108. The present value of Rs. 108 at an 8% discount rate is Rs. 100. Therefore, the call value cannot fall below Rs. 2 (102 - 100).

If a dividend is expected, the lower bound of the call value is the stock value minus the expected dividend and the present value of the strike price. For instance, if the stock trades at Rs. 50, the strike price is Rs. 20, and the expected dividend is Rs. 5, the lower bound of the call value is Rs. 26.85 (50 - 4.63 - 18.52).

The lower bounds of European call options, including the impact of dividends.

To summarize:

Lower bound of European call option = Max (0, Stock price - Present value of strike price - Present value of dividend)

In the example you provided:

• Stock price = Rs. 50

• Strike price = Rs. 20

• Dividend = Rs. 5

• Risk-free rate = 8%

Present value of strike price = Rs. 18.52

Present value of dividend = Rs. 4.63

The lower bound of the call option = Rs. 50 - (Rs. 18.52 + Rs. 4.63) = Rs. 26.85

This means that the value of the European call option cannot fall below Rs. 26.85.